Icositrigon
Regular icositrigon | |
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Dual polygon | Self |
In geometry, an icositrigon (or icosikaitrigon) or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.
Regular icositrigon
A regular icositrigon is represented by Schläfli symbol {23}.
A regular icositrigon has
The regular icositrigon is not
Concerning the nonconstructability of the regular icositrigon, A. Baragar (2002) showed it is not possible to construct a regular 23-gon using only a compass and twice-notched straightedge by demonstrating that every point constructible with said method lies in a tower of fields over such that , being a sequence of nested fields in which the degree of the extension at each step is 2, 3, 5, or 6.
Suppose in is constructible using a compass and twice-notched straightedge. Then belongs to a field that lies in a tower of fields for which the index at each step is 2, 3, 5, or 6. In particular, if , then the only primes dividing are 2, 3, and 5. (Theorem 5.1)
If we can construct the regular p-gon, then we can construct , which is the root of an irreducible polynomial of degree . By Theorem 5.1, lies in a field of degree over , where the only primes that divide are 2, 3, and 5. But is a subfield of , so divides . In particular, for , must be divisible by 11, and for , N must be divisible by 7.[2]
This result establishes, considering prime-power regular polygons less than the 100-gon, that it is impossible to construct the 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, and 89-gons with neusis. But it is not strong enough to decide the cases of the 11-, 25-, 31-, 41-, and 61-gons. Elliot Benjamin and Chip Snyder discovered in 2014 that the regular hendecagon (11-gon) is neusis constructible; the remaining cases are still open.[3]
An icositrigon is not
Related figures
Below is a table of ten regular icositrigrams, or
{23/2} |
{23/3} |
{23/4} |
{23/5} |
{23/6} |
{23/7} |
{23/8} |
{23/9} |
{23/10} |
{23/11} |
References
- OEIS; https://oeis.org/A048136
- ^ Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society 156.3 (May 2014): 409-424.; https://dx.doi.org/10.1017/S0305004113000753
- ^ Young Lee, H. (2017) Origami-Constructible Numbers University of Georgia https://getd.libs.uga.edu/pdfs/lee_hwa-young_201712_ma.pdf
- ^ P. Milici, R. Dawson The equiangular compass December 1st, 2012, The Mathematical Intelligencer, Vol. 34, Issue 4 https://www.researchgate.net/profile/Pietro_Milici2/publication/257393577_The_Equiangular_Compass/links/5d4c687da6fdcc370a8725e0/The-Equiangular-Compass.pdf